Hands down, Southwest. Great airline for kids to fly - free entertainment most of the time, in the form of Flight Attendants. We've had them recite poems, Sing, dance and one even gave my daughter a HUGE bag of Peanut bags (like 500 of them) because she was giving the flight attendant a bad time.
Also, they have real sales and fairly low fares most other times.

Tom:

That is very interesting about Southwest not flying from LAX - SFO. I didn't know that. We fly very frequently on Southwest but not to/from LAX. The vast majority of our flights are to/from San Diego. Southwest does fly to/from San Diego into SJC, OAK, and SFO. I wonder why they do not do LAX - SFO.

My guess is that Southwest opts for OAK-LAX and SJC-LAX instead of the SFO-LAX because (a) United and other carriers have already maxed out the SFO-LAX shuttle route and (b) SFO's notorious delays would really cramp the Southwest system as SW only allows 15-25 minutes between flights. Had Southwest not already been at SFO when they opened their OAK station in 88 or 89, I suspect they would have never gone there! On the east coast, I doubt we'll ever see Southwest at Logan, LaGuardia or National. They will happily (and profitably) serve Providence/Manchester, Islip/Long Island and Baltimore/Washington instead! I LUV that airline!!

Are there any others that fly to and from East Coast? S.W. doesn't fly much out of Florida,
Georgia, Alabama to NY, and Boston for example.

US Airways probably has the biggest concentration on the east coast, with AirTran, Midway, Continental, and Delta being good choices as well in terms of flights up and down the coast.

Southwest does fly from Birmingham, AL and 4 Florida cities to the Boston area (Manchester/Providence). Probably would involve a stop in Baltimore or another city, though.